Power

Boxplots

Boxplot Basics

A boxplot splits the data set into
quartiles.
The body of the boxplot consists of a "box" (hence, the name),
which goes from the first quartile (Q1) to the third quartile
(Q3).

Within the box, a vertical line is drawn at the Q2, the
median
of the data set. Two horizontal lines, called
whiskers, extend from the front and back of the box.
The front whisker goes from Q1 to the smallest non-outlier in the data set,
and the back whisker goes from Q3 to the largest non-outlier.

Smallest non-outlier

Q1

Q2

Q3

Largest non-outlier

..

-6

-4

-2

0

2

4

6

8

10

12

14

16

If the data set includes one or more
outliers, they
are plotted separately as points on the chart. In the boxplot
above, two outliers are shown to the right of
the second whisker.

How to Interpret a Boxplot

Here is how to read a boxplot. The median is indicated by the
vertical line that runs down the center of the box. In the boxplot
above, the median is between 4 and 6, around 5.

Additionally, boxplots display two common measures of the variability
or spread in a data set.

Range.
If you are interested in the spread of all the
data, it is represented on a boxplot by the horizontal
distance between the smallest value and the largest value,
including any outliers.
In the boxplot above, data values range from about 0
(the smallest non-outlier) to about 16 (the largest outlier), so
the range is 16. If you ignore outliers, the range is illustrated
by the distance between the opposite ends of the whiskers -
about 10 in the boxplot above.

Interquartile range (IQR). The middle half of a data
set falls within the interquartile range. In a boxplot,
the interquartile range is represented by the width of
the box (Q3 minus Q1). In the chart above, the interquartile
range is equal to about 7 minus 3 or about 4.

And finally, boxplots often provide information about the shape
of a data set. The examples below show some common
patterns.

2

4

6

8

10

12

14

16

Skewed right

2

4

6

8

10

12

14

16

Symmetric

2

4

6

8

10

12

14

16

Skewed left

Each of the above boxplots illustrates a different
skewness
pattern. If most of the observations are concentrated on
the low end of the scale, the distribution is skewed right;
and vice versa. If a distribution is symmetric, the
observations will be evenly split at the median, as
shown above in the middle figure.

Test Your Understanding

Problem 1

Consider the boxplot below.

2

4

6

8

10

12

14

16

18

Which of the following statements are true?

I. The distribution is skewed right.
II. The interquartile range is about 8.
III. The median is about 10.

(A) I only
(B) II only
(C) III only
(D) I and III
(E) II and III

Solution

The correct answer is (B). Most of the observations are on the
high end of the scale, so the distribution is
skewed left.
The
interquartile range is indicated by the length of the
box, which is 18 minus 10 or 8.
And the median is indicated by the vertical line running through
the middle of the box, which is roughly centered over 15. So the
median is about 15.